Step |
Hyp |
Ref |
Expression |
1 |
|
evlfval.e |
⊢ 𝐸 = ( 𝐶 evalF 𝐷 ) |
2 |
|
evlfval.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
3 |
|
evlfval.d |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
4 |
|
evlfval.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
5 |
|
evlfval.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
6 |
|
evlfval.o |
⊢ · = ( comp ‘ 𝐷 ) |
7 |
|
evlfval.n |
⊢ 𝑁 = ( 𝐶 Nat 𝐷 ) |
8 |
|
evlf2.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) |
9 |
|
evlf2.g |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) |
10 |
|
evlf2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
11 |
|
evlf2.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
12 |
|
evlf2.l |
⊢ 𝐿 = ( 〈 𝐹 , 𝑋 〉 ( 2nd ‘ 𝐸 ) 〈 𝐺 , 𝑌 〉 ) |
13 |
|
evlf2val.a |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) ) |
14 |
|
evlf2val.k |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝑋 𝐻 𝑌 ) ) |
15 |
1 2 3 4 5 6 7 8 9 10 11 12
|
evlf2 |
⊢ ( 𝜑 → 𝐿 = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑌 ) 〉 · ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) ) |
16 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑎 = 𝐴 ∧ 𝑔 = 𝐾 ) ) → 𝑎 = 𝐴 ) |
17 |
16
|
fveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑎 = 𝐴 ∧ 𝑔 = 𝐾 ) ) → ( 𝑎 ‘ 𝑌 ) = ( 𝐴 ‘ 𝑌 ) ) |
18 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑎 = 𝐴 ∧ 𝑔 = 𝐾 ) ) → 𝑔 = 𝐾 ) |
19 |
18
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑎 = 𝐴 ∧ 𝑔 = 𝐾 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) = ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝐾 ) ) |
20 |
17 19
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑎 = 𝐴 ∧ 𝑔 = 𝐾 ) ) → ( ( 𝑎 ‘ 𝑌 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑌 ) 〉 · ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) = ( ( 𝐴 ‘ 𝑌 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑌 ) 〉 · ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝐾 ) ) ) |
21 |
|
ovexd |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑌 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑌 ) 〉 · ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝐾 ) ) ∈ V ) |
22 |
15 20 13 14 21
|
ovmpod |
⊢ ( 𝜑 → ( 𝐴 𝐿 𝐾 ) = ( ( 𝐴 ‘ 𝑌 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑌 ) 〉 · ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ‘ 𝐾 ) ) ) |